V/W FORMS A VECTOR SPACE : LINEAR ALGEBRA : B.SC. MATHEMATICS #linear algebra #degree

For a video explanation, click the link 👉https://youtu.be/unnIkQ7DhuM    

THEOREM :

     Let W be a subspace of a vector space V ( F ) . Then the set V/W = { W + 𝛂 / 𝛂 𝞊 V } of 

      all cosets forms a Vector Space under  the Coset addition and Scalar multiplication

      are defined as 

      i.  For W + 𝛂 , W + 𝞫 𝞊 V,   ( W +  𝛂 ) + ( W +  𝞫 ) = W + ( 𝛂  +  𝞫 ) 

     ii . For a 𝞊  F , W + 𝛂 𝞊 V,    a  ( W +  𝛂 ) =  ( W +  a𝛂 ).

 PROOF : 

* Find the coordinates of 𝜶 = (4 ,5 ,6) w.r.t.  x=(1,1,1),y = (-1,1,1),z = (1,0,-1) 

*dim ( W1 + W2 ) = dim ( W1 ) + dim ( W2) - dim ( W1 ⋂ W2 ) 

*  If 𝜶 = ( 1 , -1 , 0 ) , 𝜷 = ( 2 , 1 , 3 ) find a basis for R3 containing 𝜶 and 𝜷 

*  Show that the set { ( 1 , 2 , 1 ) , ( 2 , 1 , 0 ) , ( 1 , -1 , 2  } forms a basis of V3 ( F )

 * Theorem on Linear Dependence 

* linear sum of subspaces

* What is a vector space

* linear span of a set 

  * L ( S ) = L ( S' )

 * Theorem on vector space  

 * Historical Introduction to Linear Algebra

 * Let V(F) be a vector space and let W ⊆ V. The necessary and sufficient conditions for W 

            to be a subspace of V are (i) 𝜶 𝞊 W, 𝞫 𝞊 W ⇒  𝜶 - 𝞫 𝞊 W

                                                     (ii) a 𝞊 F , 𝜶 𝞊 W ⇒ a𝜶 𝞊 W.

* Let V(F) be a vector space and let W ⊆ V. The necessary and sufficient conditions for W 

    to be a subspace of V is  𝜶 𝞊 W, 𝞫 𝞊 W ⇒  a𝜶 +b𝞫 𝞊 W

 * Problem 1 on subspace of a vector space 

* Problem 2 on subspace of a vector space

 * Intersection of subspaces is again a subspace

  * union of subspaces of a vector space

  * Problems on linear combination

 * Pairwise sums of Linearly Independent Vectors 

 * Quotient Set 

#V/Wformsavectorspace  #v/w #forms #a #vector #space #quotient #space 

#vector #algebra #vector algebra   #iit #jee #mains #dimension #basis #subspacec #